Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations

Fan, Lili; Ruan, Lizhi; Xiang, Wei

doi:10.1016/j.anihpc.2018.03.008

Cited by 23 publications

(11 citation statements)

References 42 publications

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“…Finally, for the case of composite waves, the stability of the composite wave of rarefaction waves and a viscous contact wave was investigated in [33,43]. Recently the authors in [4] studied the unique global-in-time existence and the asymptotic stability of the composite wave of two viscous shock waves by employing the anti-derivative method.…”

Section: Ifmentioning

confidence: 99%

“…(1. 4) In what follows, we give a roughly classification of the time-asymptotic states of the solution (ρ,u,θ,q)(x,t) based on the boundary data (ρ,u,θ,q)(0,t). It is expected that as time tends to the infinity, the solution is asymptotically described by one of the following waves, such as viscous shock wave, stationary wave, rarefaction wave or the superposition of stationary wave and rarefaction wave.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Stability of Shock Wave for the Outflow Problem Governed by the One-Dimensional Radiative Euler Equations

Fan¹

2022

CMAA

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This paper is devoted to the study of the asymptotic stability of the shock wave of the outflow problem governed by the one-dimensional radiative Euler equations, which are a fundamental system in the radiative hydrodynamics with many practical applications in astrophysical and nuclear phenomena. The outflow problem means that the flow velocity on the boundary is negative.Comparing with our previous work on the asymptotic stability of the rarefaction wave of the outflow problem for the radiative Euler equations in [6], two points should be pointed out. On one hand, boundary condition on velocity is considered instead of boundary condition on temperature, which induces a perfect boundary condition on anti-derivative perturbations so that boundary estimates on perturbed unknowns are trickily and smoothly established. On the other hand, the rarefaction wave is an expansive wave, while the shock wave is a compressive wave. So we need take good advantages of properties of the shock wave instead. Our investigation on the outflow problem provides a good understanding on the radiative effect and boundary effect in the setting of shock wave.

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Section: Ifmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Asymptotic Stability of Shock Wave for the Outflow Problem Governed by the One-Dimensional Radiative Euler Equations

Fan¹

2022

CMAA

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“…Finally, for the case of composite waves, the stability of the composite wave of rarefaction waves and a viscous contact wave was investigated in [32,40]. Recently the authors in [5] studied the unique global-in-time existence and the asymptotic stability of the composite wave of two viscous shock waves by employing the anti-derivative method.…”

Section: Related Literaturementioning

confidence: 99%

Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations

Fan

Ruan

Xiang³

2021

Discrete &Amp; Continuous Dynamical Systems - A

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This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [6], since the rarefaction wave is a nonlinear expansive wave, while the contact discontinuity wave is a linearly degenerate diffusive wave. So we need to take good advantage of properties of the viscous contact discontinuity wave instead. Moreover, series of tricky argument on the boundary is done carefully based on the construction and the properties of the viscous contact discontinuity wave for the radiative Euler equations. Our result shows that radiation contributes to the stabilization effect for the supersonic inflow problem.

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“…As in Refs. 7–9, we introduce the linear diffusion waves

false({\tilde{ψ}}^{β}, {\tilde{θ}}^{β} false) false(x, t false)

and the correct function

{\overset{θ}{true}}^{β} false(x, t false)

defined, respectively, by () and () later as the perturbation corresponding to the Cauchy problem () and () and let

({, \begin{matrix} u^{β} false(x, t false) = ψ^{β} false(x, t false) - {\overset{ψ}{true}}^{β} false(x, t false), \\ v^{β} false(x, t false) = θ^{β} false(x, t false) - {\overset{θ}{true}}^{β} false(x, t false) - {\overset{θ}{true}}^{β} false(x, t false) . \end{matrix})

By direct calculations, the Cauchy problem () and () can be rewritten as follows:

({, \begin{matrix} u_{t}^{β} & = & 0true - (1 - α) u^{β} - v_{x}^{β} + α u_{x x}^{β} - {\overset{̌}{θ}}_{x}^{β}, \\ v_{t}^{β} & = & 0true - (1 - β) v^{β} + ν u_{x}^{β} + {false(u^{β} v^{β} false)}_{x} + β v_{x x}^{β} \\ 0true + {false({\tilde{ψ}}^{β} v^{β} false)}_{x} + (...) \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

Convergence rates of vanishing diffusion limit on conservative form of Hsieh's equation

Fan

Trouba

2021

Stud Appl Math

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The aim of this paper is to study the global unique solvability on Sobolev solution perturbated around diffusion waves to the Cauchy problem of conservative form of Hsieh's equations. Furthermore, convergence rates are also obtained as one of the diffusion parameters goes to zero. The difficulty is created due to conservative nonlinearity to enclose the uniform (in diffusion parameter) higher order energy estimates. However this kind of difficulty will not occur for both the nonconservative nonlinearity and fixed diffusion parameter. The more subtle mathematical analysis needs to be introduced to overcome the difficulties.

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Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations

Cited by 23 publications

References 42 publications

Asymptotic Stability of Shock Wave for the Outflow Problem Governed by the One-Dimensional Radiative Euler Equations

Asymptotic Stability of Shock Wave for the Outflow Problem Governed by the One-Dimensional Radiative Euler Equations

Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations

Convergence rates of vanishing diffusion limit on conservative form of Hsieh's equation

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